Let $X_n, n\in\mathbb{Z}$ be a sequence of independent random variables (finite, let's say the size of the alphabet is $2$ to simplify things) with mean zero and variance less than $1$. Is there a large deviations type result for this case? i.e. Does $P\left[|\frac{X_1+\cdots+X_n}{n}|\ge t\right]$ fall exponentially?
The motivation is this: Let $(X_n,Y_n),n\in \mathbb{Z}$ be a sequence of iid random variables with joint distribution $p_{XY}=p_YW_{X|Y}$ for some transition kernel $W_{X|Y}$. Can we state a "Conditional Large Deviations" result for $X^n$ given the outcome $Y^n=y^n$ where $y^n$ is a likely outcome of $Y^n$ (Let's say the empirical distribution of $y^n$ is close to $P_Y$ with respect to some metric) Here $Y$ can take values from $\mathbb{R}$ but $X$ is finite.
One can realize each $X_n$ as $\rho(Y_n,Z_n)$ for some i.i.d. sequence $(Z_n)_n$ independent of $(Y_n)_n$. Then the result holds (in a somewhat greater generality) for almost every realization of $(Y_n)_n$, see Stochastic Sub-Additivity Approach to the Conditional Large Deviation Principle by Zhiyi Chi.